Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations

We consider the numerical discretization of the time-domain Maxwell's equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach.Comment: 16 pages, 1 figure, 1 table; Springer Wien, ISBN 978-3-7091-0793-

A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented

Higher-Order Spectral/HP Finite Element Technology for Structures and Fluid Flows

This study deals with the use of high-order spectral/hp approximation functions in the finite element models of various nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order p greater than or equal to 4) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., p < 3) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements for beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. The higher-order spectral/hp basis functions avoid the interpolation error in the numerical schemes, thereby making them accurate and stable. Furthermore, for fluid flows, when combined with least-squares variational principles, such technology allows us to develop efficient finite element models, that always yield a symmetric positive-definite (SPD) coefficient matrix, and thereby robust direct or iterative solvers can be used. The least-squares formulation avoids ad-hoc stabilization methods employed with traditional low-order weak-form Galerkin formulations. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities (e.g., dilatation, volume, and mass) and stable evolution of variables with time in the case of unsteady flows. The present study uses spectral/hp approximations in the (1) weak-form Galerkin finite element models of viscoelastic beams, (2) weak-form Galerkin displacement finite element models of shear-deformable elastic shell structures under thermal and mechanical loads, and (3) least-squares formulations for the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical simulations using the developed technology of several non-trivial benchmark problems are presented to illustrate the robustness of the higher-order spectral/hp based finite element technology

Hybrid multigrid methods for high-order discontinuous Galerkin discretizations

The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the SIPG method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms

Application of the Fekete spectral element method in stress analysis of plates involving holes and complex geometries

In an effort to prove the effectiveness of a methodology that computes more efficiently than traditional FEM methods, this paper details a method that allows for the accurate and efficient solution of an elasticity problem using an hp-SEM code. The code can interchange basis functions to allow us to compare Lagrange with Fekete basis functions, which have the capability of high accuracy and efficiency. The code is applied to solve a traditional elasticity problem with an analytical solution to judge the accuracy on a course mesh. This gives us a method that provides greater accuracy than traditional FEM when comparing the same mesh, and higher efficiency compared to Lagrange bases comparing matrix condition numbers. This produces a flatter curve when varying p-order, which shows that higher orders than the 9th-order methods tested here are easily achievable with this technique. The flexibility of our code is shown by solving solutions with complex geometry and holes